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The two routes are typically developed in isolation. We make a structural observation about KL-Shampoo's Kronecker preconditioners: their eigenvalue spectra exhibit a \\emph{spike-and-flat} shape -- a few dominant eigenvalues followed by an approximately uniform tail -- across layers and training stages, holding exactly under a rank-$\\rho$ signal-plus-noise gradient model. We exploit this structure by restricting one of KL-Shampoo's Kronecker factors to a parametric family aligned with the spike-and-flat shape: full spectral structure on a tracked $","title":"Pro-KLShampoo: Projected KL-Shampoo with Whitening Recovered by Orthogonalization","url":"https://arxiv.org/abs/2605.06316","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.06316v1 Announce Type: cross \nAbstract: Optimizers that exploit the matrix structure of gradients are central to modern LLM pre-training, with two distinct frontiers: explicit Kronecker-factored preconditioning -- most recently KL-Shampoo, which estimates the preconditioner via KL divergence minimization -- and orthogonalization of the gradient momentum, exemplified by Muon and analyzed as steepest descent under the spectral norm. The two routes are typically developed in isolation. 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We exploit this structure by restricting one of KL-Shampoo's Kronecker factors to a parametric family aligned with the spike-and-flat shape: full spectral structure on a tracked $","title":"Pro-KLShampoo: Projected KL-Shampoo with Whitening Recovered by Orthogonalization","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-08T04:43:40Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.06316"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:f19a9d40c15f07de1bcadbf659fb44095a8481b58777c7e9ffb1747e5c75b2b9b13ed452fc9e7ccfb06823da375b53ac0eae7790c491e2af547ff84c8cfbce00","signer":"crovia.substrate","subject":{"observed_at":"2026-05-08T04:43:40Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.06316"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"6c5e6f34dcb21fd2713c51f982c67ba5f5a2f778ed418b87a65ad8219e70e700","leaf_index":120305,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"b2a720c26f83158b6d145c8fda6f21a4a4ad7f00e1ca612bd00897d2a0ec59c0","side":"left"},{"sibling":"9d88c1250daac2e4039711b72cbff1fa411a02c7060a90124518d9f71d2909de","side":"right"},{"sibling":"544d00c5187466a8e86d4e630544c682d6af766e2d3e2eec244fbee1790f323c","side":"right"},{"sibling":"179614c60478ccad43e9e95839355a4144ec5386b9862252140e11f0a9ff986f","side":"right"},{"sibling":"7ce3b1011b65acc0647040aec43a789ccd5ba0750d554972572310281ad373ca","side":"left"},{"sibling":"7d725f913e30cfa2dc7f0dfc239f94e99329ac388d82deaf401b622d087e16d8","side":"left"},{"sibling":"f8731f97e8f9ae56bf080fa7b1f75b46c0f0aa51627cbe9299dc94fe90595ade","side":"left"},{"sibling":"6f9fbeeae57a07a1aca1b75e07f4ae2c8f837e9a25cd31813866783835b2d912","side":"left"},{"sibling":"34e93c6f591cc4fb93a4c29771210eb73573b454d2c5329963f437f1309fe46f","side":"left"},{"sibling":"b9834433f5bd1deeaab4ced2b3bcc0d91d19763a5d0981fb299d124b63459eb3","side":"right"},{"sibling":"143f33d3924b3840fd6dd8ba12566fc35bf86189e663ef0ad4884d676f295e3a","side":"left"},{"sibling":"b1ed99341c327c7c9ab2489f40af2547ab3b3b4b6a74fb684210164fb891a413","side":"right"},{"sibling":"6ee3be9bdfc9bee55d32f7dbb0075f02fe87d20887d563d3e300caf36b1b88c7","side":"left"},{"sibling":"8ccd9937a2c0d5c04044d07d1557791b7d07bb31eac41a39a675608d44b38f23","side":"right"},{"sibling":"3a5e69cf0803f4c91f3895ed7c9a95748fef240bec4422e167c05300f79f06c0","side":"left"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_ac8c7f7da870ca9505a48fe35f106fa72a0884807a660d43d361a758ece3a1f6"}}